Nuclear magnetic resonance (NMR) is a useful tool in the determination of the pore size distribution (PSD) of a porous medium. The PSD of a medium can be determined by making relaxation measurements of a fluid saturating the medium. In particular, the T2 spin-spin relaxation time is related to pore size according to
            1              T        2              =                  1                  T                      2            ⁢            b                              +              ρ        ⁢                  S          V                      ,where T2b is the T2 value of the bulk fluid, ρ is the surface relaxivity, and S/V is the surface volume ratio. See, e.g., Kleinberg, R., et al., “Utility of NMR T2 distributions, connection with capillary pressure, clay effect, and determination of the surface relaxivity parameter ρ2,” Magnetic Resonance Imaging, Vol. 14 pp. 761-767 (1996)). The determination of surface relaxivity ρ may be considered important for quantitative interpretation of NMR data. Surface relaxivity is usually determined by finding the value of ρ that matches the PSD derived from the NMR T2 distribution to the PSD of an independent laboratory measurement, for example via a capillary pressure measurement. See, e.g., Borgia, G., et al., “Nuclear magnetic resonance relaxivity and surface-to-volume ratio in porous media with a wide distribution of pore sizes,” Journal of Applied Physics, Vol. 79 pp. 3656-3664 (1996); and Basan, P. B., et al., “Pore-size data in petrophysics: a perspective on the measurement of pore geometry,” Geological Society, London, Special Publications, Vol. 122 pp. 47-67 (1997). However, an independent measurement of the surface area is not always available, such as in formation (borehole) logging applications where the medium remains in situ.
When water is located inside a porous medium, motion of the water is affected by the solid matrix of that medium. The corresponding effect on the apparent diffusion coefficient is well established both theoretically and experimentally and that effect may be utilized to probe the S/V information. See, Mitra, P. P., et al., “Diffusion propagator as a probe of the structure of porous media,” Physical Review Letters, Vol. 68 pp. 3555-3558 (1992); and Latour, L. L., et al., “Time-dependent diffusion coefficient of fluids in porous media as a probe of surface-to-volume ratio,” Journal of Magnetic Resonance, Series A, Vol. 101 pp. 342-346 (1993). Recently, a purely in-situ NMR logging approach that combines T2 relaxation and diffusion measurements to determine surface relaxivity was proposed. See, Zielinski, L., et al., “Restricted Diffusion Effects in Saturation Estimates from 2D Diffusion-Relaxation NMR Maps,” SPE Annual Technical Conference and Exhibition, 2010; and Zielinski, L., et al., “Method for Determining Rock Formation Fluid Interaction Properties Using Nuclear Magnetic Resonance Well Logging Measurements,” U.S. Patent Publication #2013/0057277 (2013) which is hereby incorporated by reference herein in its entirety. An apparent advantage of using restricted diffusion to determine S/V is that T2 relaxation is affected by paramagnetic centers on the surface through the diffusion process. See, Brownstein, K. R., and Tarr, C., “Importance of classicial diffusion in NMR studies of water in biological cells,” Physical Review A, Vol. 19 p. 2446 (1979); Wilkinson, D. J., et al., “Nuclear magnetic relaxation in porous media: The role of the mean lifetime τ(ρ,D),” Physical Review B, Vol. 44 p. 4960 (1991). Unlike systems that use one dimensional time dependent diffusion data to determine surface relaxivity, Zielinski et al. analyze a 2D diffusion-relaxation (DT2) map which contains S/V information as function of pore size instead of the ratio of total surface to total volume (ST/VT).
Zielinski et al.'s methodology is suitable to homogenous (e.g., single pore size length scale) sample applications where T2 differs notably from its bulk value. The methodology utilizes inputs to the analysis such as tortuosity, bulk diffusion coefficients, and a fixed heterogeneity length.